Quadratic penalty method ($f(x)+\frac{1}{2\mu }(\Vert h(x){\Vert }^2+\Vert g(x{)}^{-}{\Vert }^2)$)

For constraints $h=0$, $g\ge 0$, minimize the displayed merit function for progressively smaller $\mu >0$, where $g_i^{-}=\min (g_i,0)$. Increasing the quadratic penalty discourages constraint violation. In a violated inequality, differentiation gives the multiplier estimate ${\lambda }_i=-g_i/\mu$; an inactive inequality has estimate zero. Thus obtaining small violations by this method alone typically requires a very small $\mu$, which may cause a large condition number in the inner minimization.